A130780 - OEIS

%I #24 Dec 12 2021 11:19:45

%S 1,1,1,3,3,6,8,12,16,23,32,42,58,75,102,131,173,220,288,363,466,587,

%T 743,929,1164,1448,1797,2224,2738,3368,4122,5042,6133,7466,9035,10941,

%U 13184,15888,19064,22876,27343

%N Number of partitions of n such that number of odd parts is greater than or equal to number of even parts.

%C a(n) = A108950(n) + A045931(n) = A000041(n) - A108949(n). - _Reinhard Zumkeller_, Jan 21 2010

%C a(n) = Sum_{k=0..n} A240009(n,k). - _Alois P. Heinz_, Mar 30 2014

%H Alois P. Heinz, <a href="/A130780/b130780.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: Sum_{k>=0} x^k/Product_{i=1..k} (1-x^(2*i))^2.

%e a(5)=6 because we have 5,41,32,311,211 and 11111 (221 does not qualify).

%p g:=sum(x^k/(product((1-x^(2*i))^2,i=1..k)),k=0..50): gser:=series(g,x=0,50): seq(coeff(gser,x,n),n=1..40); # _Emeric Deutsch_, Aug 24 2007

%p # second Maple program:

%p b:= proc(n, i, t) option remember; `if`(n=0,

%p       `if`(t>=0, 1, 0), `if`(i<1, 0, b(n, i-1, t)+

%p       `if`(i>n, 0, b(n-i, i, t+(2*irem(i, 2)-1)))))

%p     end:

%p a:= n-> b(n$2, 0):

%p seq(a(n), n=0..80);  # _Alois P. Heinz_, Mar 30 2014

%t $RecursionLimit = 1000; b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[t >= 0, 1, 0],  If[i<1, 0, b[n, i-1, t] + If[i>n, 0, b[n-i, i, t + (2*Mod[i, 2]-1)]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 80}] (* _Jean-François Alcover_, May 12 2015, after _Alois P. Heinz_ *)

%t opgQ[n_]:=Module[{len=Length[n],op},op=Length[Select[n,OddQ]];op>= len-op]; Table[Count[IntegerPartitions[n],_?(opgQ)],{n,0,50}] (* _Harvey P. Dale_, Dec 12 2021 *)

%Y Cf. A045931, A108949, A108950.

%Y Cf. A171966, A171967. - _Reinhard Zumkeller_, Jan 21 2010

%K easy,nonn

%O 0,4

%A _Vladeta Jovovic_, Aug 19 2007

%E More terms from _Emeric Deutsch_, Aug 24 2007